Ingeniería Biomédica
2025-07-08
Even
\[f\left(t\right) = f\left(-t\right)\] \[f\left[t\right] = f\left[-t\right]\]
Odd
\[f\left(t\right) = -f\left(-t\right)\] \[f\left[t\right] = -f\left[-t\right]\]
Decomposition
All signal can be decomposed in two signals: one even, one odd.
\[x(t) = x_{even}(t) + x_{odd}(t)\]
Where:
\[x_{even}(t) = \frac{x(t)+x(-t)}{2} \] \[x_{odd}(t) = \frac{x(t)-x(-t)}{2} \]
Example
Decompose the signal \(x(t)=e^{t}\) into its even and odd parts
\[x_{\text{even}}(t) = \frac{x(t) + x(-t)}{2}\]
\[x_{\text{odd}}(t) = \frac{x(t) - x(-t)}{2}\]
\[x(-t) = e^{-t}\]
\[x_{\text{even}}(t) = \frac{e^t + e^{-t}}{2} = \cosh(t)\]
\[x_{\text{odd}}(t) = \frac{e^t - e^{-t}}{2} = \sinh(t)\]
\[x(t) = x_{\text{even}}(t) + x_{\text{odd}}(t)\]
\[e^t = \cosh(t) + \sinh(t)\]
Signals can undergo two types of transformations:
Consider: [ y(t) = 2 x(3t - 1) + 1 ] 1. Time compression: ( x(3t) ) compresses the signal. 2. Time shift: ( x(3t - 1) ) shifts it to the right by 1 unit. 3. Amplitude scaling: ( 2 x(3t - 1) ) amplifies the signal. 4. Amplitude shift: ( +1 ) shifts it upward.